CHARACTERIZATION OF ABELIAN VARIETIES Jungkai A. Chen, Christopher D. Hacon

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Jungkai A. Chen, Christopher D. Hacon

Abstract. We prove that any smooth complex projective variety X with plurigenera P1(X) = P2(X) = 1 and irregularity q(X) = dim(X) is birational to an abelian variety.

Introduction

Given a smooth projective variety X, one would like to characterize it by its birational invariants. Classical results include Castelnuovo’s criterion for rational surfaces, which states that a complex surface with plurigenera P2(X) = 0 and irregularity q(X) = 0 must be rational. Another well known result due to Enriques, characterizes abelian surfaces as the only surfaces with P1(X) = P2(X) = 1 and q(X) = 2.

In an attempt to generalize Enriques’ result to higher dimensional varieties, Kawamata proved that a variety X with Kodaira dimension κ(X) = 0 and irregularity q(X) = dim(X) admits a birational map to an abelian variety [Ka1]. In [Ko1], Koll´ar gives an effective characterization by proving that if P1(X) = P4(X) = 1 and X has maximal Albanese dimension, then X is birational to an abelian variety.

Ein and Lazarsfeld prove a similar result by using generic vanishing theorems [EL1].

Koll´ar subsequently improved his result by weakening the hypothesis to P3(X) = 1 and q(X) = dim(X). He conjectured that the result still holds if one assumes that P2(X) = 1 and X is generically finite over an abelian variety [Ko2].

In this paper, we verify Koll´ar’s conjecture. Our method is based on the ap- proaches of Koll´ar and of Ein and Lazarsfeld. We would like to emphasize that the above mentioned result of Kawamata and a theorem of C. Simpson (cf. Theorem 1.3) are essential steps in the proof.

Acknowledgment. We are in debt to R. Lazarsfeld, L. Ein, Y. Kawamata, and J. Koll´ar for valuable conversations during the preparation of this paper. We would also like to thank the referee for the helpful comments.

Conventions and Notations

(0.1) Throughout this paper, we work over the field of complex numbers C.

(0.2) For D1, D2 Q-divisors on a variety X, we write D1 ≺ D2 if D2− D1 is effective, and D1≡ D2 if D1 and D2 are numerically equivalent.

The first author was partially supported by National Science Council (NSC-87-2119-M-194- 007)

1991 Mathematics Subject Classification. Primary 14H45, 14H99; Secondary 14H50.

Typeset by AMS-TEX

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(0.3) |D| will denote the linear series associated to the divisor D, and Bs|D|

denotes the base locus of |D|.

(0.4) For a real number a, let bac be the largest integer ≤ a and dae be the smallest integer ≥ a. For a Q-divisor D = P aiD_i, let bDc = PbaicDi and dDe =PdaieD_i.

(0.5) Let F be a coherent sheaf on X, then hⁱ(X, F ) denotes the complex dimension of Hⁱ(X, F ). In particular, the plurigenera h⁰(X, ω_X^⊗m) are denoted by Pm(X) and the irregularity h⁰(X, Ω¹_X) is denoted by q(X).

(0.6) Let A be an abelian variety, for all positive integers n, we will denote by An

the set of n-division points of A. For all subgroups T of A, T₀will be the connected component of T containing the origin. For any line bundle L there is a canonical homomorphism φ_L: A −→ Pic⁰(A) defined by x −→ t^∗_xL ⊗ L⁻¹. Denote by K(L) the kernel of φ_L, and by p_L: A −→ A(L) := A/K(L)₀ the induced quotient map.

1. Cohomological Support Loci

We start by recalling some results about cohomological support loci that will be needed in this paper. Most of these results may be found in [EL1] and [EV].

Let f : X → A be a morphism from a smooth projective variety X to an abelian variety A. If F is a coherent sheaf on X, then one can define the cohomological support loci by

Definition 1.1.

Vi(X, A, F ) := {P ∈ Pic⁰(A)|hⁱ(X, F ⊗ f^∗P ) 6= 0}.

In particular, if f = alb_X: X → Alb(X), then we simply write Vi(X, F ) := {P ∈ Pic⁰(X)|hⁱ(X, F ⊗ P ) 6= 0}.

We say that X has maximal Albanese dimension if dim(albX(X)) = dim(X).

The loci Vi(X, ωX) defined above are very useful in the study of irregular varieties and in particular of varieties of maximal Albanese dimension. The geometry of these loci is governed by the following:

Theorem 1.2 ([GL1],[GL2]).

a. Any irreducible component of V_i(X, ω_X) is a translate of a sub-torus of Pic⁰(X) and is of codimension at least i − (dim(X) − dim(albX(X))).

b. Let P be a general point of an irreducible component T of V_i(X, ω_X). Suppose that v ∈ H¹(X, OX) ∼= TPPic⁰(X) is not tangent to T . Then the sequence

Hⁱ⁻¹(X, ωX⊗ P )−→ H^∪v ⁱ(X, ωX⊗ P )−→ H^∪v ⁱ⁺¹(X, ωX⊗ P ) is exact. If v is tangent to T , then the maps in the above sequence vanish.

c. If X is a variety of maximal Albanese dimension, then

Pic⁰(X) ⊃ V0(X, ωX) ⊃ V1(X, ωX) ⊃ ... ⊃ Vn(X, ωX) = {OX}.

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Theorem 1.3 ([S]). Under the above hypothesis, every irreducible component of V_i(X, ω_X) is a translate of a sub-torus of Pic⁰(X) by a torsion point.

Remark 1. The results listed above also hold when we consider an abelian variety A and a morphism f : X → A with dimf (X) = dim(X) and we replace Pic⁰(X) by Pic⁰(A).

Proposition 1.4 [EL1, Proposition 2.1]. If P1(X) = P2(X) = 1, then the origin is an isolated point of V0(X, ωX).

Proposition 1.5 [EL1, Proposition 2.2]. If the origin is an isolated point of V0(X, ωX), then the Albanese mapping albX : X −→ Alb(X) is surjective.

Corollary 1.6 [EL1]. Let D be a reduced irreducible divisor of an abelian variety A. Then there exists a positive dimensional sub-group TD⊂ Pic⁰(X) such that for any desingularization ν : ˜D −→ D, h⁰( ˜D, ωD˜ ⊗ ν^∗P ) > 0 for all P ∈ TD.

If X has maximal Albanese dimension, then X admits a generically finite map to an abelian variety A. It is easy to see that P1(X) ≥ 1 by pulling back sections of Ωⁿ_A, where n = dim(X). Assume furthermore that P2(X) = 1, then necessarily P1(X) = 1, and so the Albanese map is surjective. In particular q(X) = dim(X).

Therefore one has:

Corollary 1.7. The following are equivalent.

a. P1(X) = P2(X) = 1, and q(X) = dim(X).

b. P2(X) = 1 and X has maximal Albanese dimension.

Next we illustrate some examples in which the geometry of X can be recovered from information on the loci Vi(X, ωX).

Theorem 1.8 ([EL2]). If X is a variety with maximal Albanese dimension and dimV0(X, ωX) = 0. Then X is birational to an abelian variety.

Proof. Let a : X → A be the Albanese map. Since P1(X) ≥ 1, the origin is an isolated point of V0(X, ωX). By Proposition 1.5, a is surjective and generically finite. By Theorem 1.2, for any P ∈ Pic⁰(A) and v ∈ H¹(A, O_A), the complex D(v)

. . . Hⁱ⁻¹(A, a_∗ω_X⊗ P )−→ H^∪v ⁱ(A, a_∗ω_X⊗ P )−→ H^∪v ⁱ⁺¹(A, a_∗ω_X⊗ P ) . . . is exact. Let P = P(H¹(A, O_A)^∗), and define the complex of vector bundles G_P^• by

G_Pⁱ = Hⁱ(A, a_∗ω_X⊗ P ) ⊗ O_P(−n + i).

Since the complex D(v) is exact for any v ∈ H¹(A, O_A), it follows that the complex G_P^• is an exact sequence of vector bundles. Let n = dim(X). An easy computation shows that

H⁰(A, a_∗ω_X⊗ P ) ⊗ Hⁿ⁻¹(P, OP(−n)) ∼= Hⁿ(A, a_∗ω_X⊗ P ) ⊗ H⁰(P, OP).

However, hⁿ(A, a_∗ω_X⊗ P ) = hⁿ(X, ω_X⊗ a^∗P ) = h⁰(X, O_X⊗ a^∗P ) and this is zero unless P = O_A. It follows that V₀(X, ω_X) consists of the origin only and that P₁(X) = 1.

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Let G^• = G_O^•

A. Define now the complex of vector bundles F^• by Fⁱ= Hⁱ(A, OA) ⊗ O_P(−n + i).

Let i : OA−→ a_∗ωX be the injection of sheaves corresponding to a nonzero section of a_∗ωX. Since h⁰(A, a_∗ωX) = 1 this is unique up to multiplication by non-zero scalar. There is a corresponding map of complexes i^• : F^• −→ G^•. Tensoring by O_P(n) and taking global sections, this induces an isomorphism H⁰(A, OA) ∼= H⁰(A, a∗ωX). Proceeding by induction (tensoring by O_P(n − r) and taking global sections), we obtain isomorphisms

H^r(A, O_A) ∼= H^r(A, a_∗ω_X).

Let Q be the cokernel of OA ,→ a_∗ωX. The induced maps Hⁱ(A, OA⊗ P ) −→

Hⁱ(A, a∗ωX⊗ P ) are all isomorphisms for all P ∈ Pic⁰(A). Therefore, hⁱ(A, Q ⊗ P ) = 0 for all i and all P ∈ Pic⁰(A). By [M], Q = 0 and thus OA ∼= a∗ωX. Therefore, a : X → A is birational.

Corollary 1.9 ([EL2]). Let X be a variety of maximal Albanese dimension. If X is not birational to an abelian variety. Then there exists a torsion line bundle Q ∈ Pic⁰(X) and a positive dimensional subgroup T of Pic⁰(X), such that h⁰(X, ω^⊗2_X ⊗ Q^⊗2 ⊗ P ) ≥ 2 for all P ∈ T . In particular, if V₀(X, ω_X) contains a positive dimensional component through the origin OX (or through any 2-torsion point P ∈ Pic⁰(X)2), then P2(X) ≥ 2.

Proof. Let TQ be any positive dimensional component of V⁰(X, ωX) containing Q ∈ Pic⁰(X) and let T = TQ− Q. Since T is a subgroup of Pic⁰(X), one sees that

−T = T . Consider the natural map

H⁰(X, ωX⊗ Q ⊗ Py) ⊗ H⁰(X, ωX⊗ Q ⊗ P_y^∗) → H⁰(X, ω^⊗2_X ⊗ Q^⊗2) and let Py vary in T . Both H⁰(X, ωX⊗ Q ⊗ Py) and H⁰(X, ωX⊗ Q ⊗ P_y^∗) are non-zero. The natural map above gives rise to a map of linear series

|KX+ Q + Py| × |KX+ Q − Py| → |2KX+ 2Q|.

Since there are infinitely many P_y ⊂ S, |2KX + 2Q| cannot consist of only one element. Therefore h⁰(X, ω^⊗2_X ⊗ Q^⊗2) ≥ 2.

We end this section by recalling a few results that will frequently be used in what follows.

Theorem 1.10 [EV2, Ka2, Ko1, ]. Let f : X −→ Y be a surjective morphism from a smooth projective variety X to a normal variety Y . Let L be a line bundle on X such that L ≡ f^∗M + ∆, where M is a Q-divisor on Y and (X, ∆) is klt. Then

a. R^jf_∗(ω_X⊗ L) is torsion free for j ≥ 0.

b. Assume furthermore that M is nef and big. Then Hⁱ(Y, R^jf_∗(ω_X⊗ L)) = 0 for i > 0, j ≥ 0.

c. Let D be a effective divisor on X such that f (D) 6= Y . Then H^j(X, ωX⊗ L) → H^j(X, ωX⊗ L(D)) is injective.

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Theorem 1.11 [Ko2, 14.7]. Let f : X −→ Y be a surjective morphism from a smooth projective variety X to a smooth variety Y , birational to an abelian variety.

Let L be a nef and big Q-divisor on Y , ∆ be a Q-divisor with normal crossings such that b∆c = 0. Let U ⊂ Y be a dense open set, and N a Cartier divisor on X, N ≡ KX+ ∆ + f^∗L. Assume that N |_g⁻¹_U is linearly equivalent to an effective divisor. Then h⁰(X, N ) 6=0.

2. Birational invariants

In this section we study birational invariants of varieties with maximal Albanese dimension. Let f : X → Y be a birational model of the Iitaka fibration of X. We may assume that X and Y are smooth projective varieties. Suppose that there is an abelian variety A and a generically finite map a : X −→ A. Let K be the image of the generic fiber X_y of f via the map a : X −→ A. Since κ(X_y) = 0, by [Ka1], it follows that X_y−→ K is ´etale in codimension 1. Moreover, K is a fixed abelian subvariety of A, and there is a generically finite map Y −→ Alb(X)/K. Thus Y is also of maximal Albanese dimension. Let D be an non-exceptional irreducible component of the ramification divisor with respect to the map a : X → A. Then P1(D) ≥ 1, and f (D) 6= Y .

Lemma 2.1. Suppose that X has maximal Albanese dimension, and positive Ko- daira dimension κ(X) > 0. Fix Q a torsion element of Pic⁰(X). Then h⁰(X, ω_X^⊗2⊗ Q ⊗ f^∗P ) is constant for all torsion P ∈ Pic⁰(Y ).

Replacing X by an appropriate birational model, we may assume that both the linear system |m1KX| and |m2KX− f^∗H| have base locus of pure codimension 1.

For general memebers B₁∈ |m1K_X| and B2∈ |m2K_X− f^∗H|, we may write

B1= D1+

r

X

i=1

aiFi, B2= D2+

r

X

i=1

biFi,

where Pr

i=1aiFi and Pr

i=1biFi denote the base divisors respectively. One can arrange that D₁6= D2 and D_i 6= Fj for all i = 1, 2 and j = 1, ..., r. Blowing up X, we may assume furthermore that Supp(B₁) ∪ Supp(B₂) is normal crossing.

Consider B = kB₁+ B₂∈ |(km1+ m₂)K_X− f^∗H|. We claim that for k 0

b B

km1+ m2

c ≺ 3 2m1

r

X

i=1

a_iF_i and

2 m1

r

X

i=1

aiFi≺ F2,Q+f^∗P, 2 m1

r

X

i=1

aiFi≺ F2,Q.

The second inequality is easily seen since ^m₂¹|2KX + Q + f^∗P | ⊂ |m1KX|. It follows thatPr

i=1a_iF_i≺ ^m₂¹F_2,Q+f^∗_P. The third inequality holds similarly.

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For the first inequality, we proceed by comparing the multiplicities of each component. Recall that B = kD1+ D2+Pr

i=1(kai+ bi)Fi. First of all, b_km^k

1+m₂c = b_km¹

1+m₂c = 0. Moreover, if ai= 0 then b ka_i+ b_i

km1+ m2

c = b b_i km1+ m2

c = 0 ≤ 3a_i 2m1

, for all k 0. If ai> 0, then

b kai+ bi

km₁+ m₂c ≤ kai+ bi

km₁+ m₂ ≤ 3ai

2m₁, for all k 0. Hence the claim holds.

Now let m = km1+ m2, and define M := KX(−bB

mc) ≡ f^∗(H m) + {B

m}.

By comparing fixed loci as above, one sees that h⁰(X, ωX(M ) ⊗ Q ⊗ f^∗P ) = h⁰(X, ω^⊗2_X ⊗Q⊗f^∗P ), and h⁰(X, ωX(M )⊗Q) = h⁰(X, ω_X^⊗2⊗Q). Since the support of B has only normal crossing singularities, {^B_m} is Kawamata log terminal. By Theorem 1.10, it follows that Hⁱ(Y, R^jf_∗(ωX(M ) ⊗ Q) ⊗ P ) = 0 for all i > 0, j ≥ 0 and all P ∈ Pic⁰(Y ). In particular

h⁰(Y, f∗(ωX(M ) ⊗ Q) ⊗ P ) = χ(Y, f∗(ωX(M ) ⊗ Q) ⊗ P ),

which is deformation invariant. So, the quantity h⁰(X, ωX(M )⊗Q⊗f^∗P ) is positive and constant for all P ∈ Pic⁰(Y ). Therefore, for any torsion element P of Pic⁰(Y ),

h⁰(X, ω^⊗2_X ⊗ Q ⊗ f^∗P ) = h⁰(X, ωX(M ) ⊗ Q ⊗ f^∗P ) = h⁰(X, ωX(M ) ⊗ Q) = h⁰(X, ω^⊗2_X ⊗ Q). An immediate consequence of Lemma 2.1 is the following

Theorem 2.2. Let X be a smooth projective variety with maximal Albanese dimension. If X is of general type, then P2(X) = h⁰(X, ω^⊗2_X ) ≥ 2.

Proof. Since X is not birational to an abelian variety, by Corollary 1.9, there exists a torsion line bundle Q ∈ Pic⁰(X), such that h⁰(X, ω_X^⊗2⊗ Q^⊗2) ≥ 2. By Lemma 2.1, h⁰(X, ω_X^⊗2) ≥ 2.

Lemma 2.3. Let X be a smooth projective variety with maximal Albanese dimension, and positive Kodaira dimension κ(X) > 0. Let f : X → Y be a birational model of the Iitaka fibration of X. Fix a canonical divisor KX and let K⁰= (KX)red. If Q is a torsion element of Pic⁰(X) such that h⁰(X, ωX⊗ Q) > 0.

Then h⁰(X, ω_X(K⁰) ⊗ Q ⊗ f^∗P ) > 0 for all P ∈ Pic⁰(Y ).

Proof. The proof proceeds as in Lemma 2.1. Let K⁰ =P Fi. For general memebers B1∈ |m1K⁰| and B2∈ |m2K⁰− f^∗H|, we may write

B1= D1+X

aiFi, B2= D2+X biFi,

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where P aiFi and P biFi denote the base divisors respectively. One can arrange that D1 6= D2 and Di 6= Fj for all i = 1, 2 and j. Blowing up X, we may assume furthermore that Supp(B1) ∪ Supp(B2) has normal crossing support.

Consider B = kB1+ B2 ∈ |(km1+ m2)K⁰− f^∗H|. We proceed by comparing the multiplicities of each component.

Clearly, m1K⁰ ∈ |m1K⁰|. Thus ai ≤ m1, for all i. One can easily check that for k 0, b_km^kaⁱ^+bⁱ

1+m2c ≤ 1.

It follows that

bB

mc ≺X

Fi= K⁰. Let

M := K⁰− bB mc.

Then h⁰(X, ω_X(M ) ⊗ Q) > 0. As in the proof of Lemma 2.1, h⁰(X, ω_X(M ) ⊗ Q ⊗ f^∗P )) is constant for all P ∈ Pic⁰(Y ) . Thus h⁰(X, ωX(M ) ⊗ Q ⊗ f^∗P ) > 0 for all P ∈ Pic⁰(Y ). It follows that h⁰(X, ω_X(K⁰) ⊗ f^∗P ) > 0 for all P ∈ Pic⁰(Y ). Theorem 2.4. Let X be a smooth projective variety with maximal Albanese dimension and positive Kodaira dimension κ(X) > 0. Let f : X → Y be the Iitaka fibration. If Y is birational to an abelian variety, then P₂(X) ≥ 2.

Proof. Let q : Y → S be a birational morphism to an abelian variety S. We identify Pic⁰(Y ) with Pic⁰(S). In what follows, we assume that h⁰(X, ω_X^⊗2) = 1 and derive a contradiction.

Step 1. Let K⁰ = (KX)red. By Lemma 2.3, h⁰(X, ωX(K⁰) ⊗ f^∗P ) ≥ 1 for all P ∈ Pic⁰(Y ). As h⁰(X, ω^⊗2_X ) = 1, equality holds for all P in an appropriate neighborhood U ⊂ Pic⁰(Y ) of the origin OY.

Since X is not birational to an abelian variety, by Corollary 1.9, there is a torsion line bundle Q in a positive dimensional component T of V0(X, ωX) such that h⁰(X, ωX⊗ Q) ≥ 1. If T ⊂ Pic⁰(Y ), then h⁰(X, ω_X^⊗2⊗ Q^⊗2) ≥ 2 by Corollary 1.9, and by Lemma 2.1, h⁰(X, ω_X^⊗2) ≥ 2. Therefore we may assume that Q 6∈ Pic⁰(Y ).

By Lemma 2.3 and its proof, h⁰(X, ω_X(M ) ⊗ Q ⊗ f^∗P ) = c ≥ 1 for all torsion P ∈ Pic⁰(Y ).

Step 2. We claim that there is a theta divisor Θ on S and an effective divisor G such that

|KX+ K⁰+ f^∗P | = f^∗q^∗|Θ + P | + G, for all P ∈ U .

To this end, let M be a divisor chosen as in the proof of Lemma 2.3 (in particular b_m^Bc ≺ K⁰). We wish to compute the cohomology groups of the direct image sheaf q∗f∗(KX + M ). From the Leray spectral sequence associated to the morphism q : Y −→ S, one sees that for all P ∈ Pic⁰(Y ),

h⁰(X, ωX⊗ M ⊗ f^∗P ) = h⁰(S, q_∗f_∗(ωX⊗ M ) ⊗ P ) = 1, and there is an injection

Hⁱ(S, q_∗f_∗(ωX⊗ M ) ⊗ P ) ,→ Hⁱ(Y, f_∗(ωX⊗ M ) ⊗ P ).

It follows that for all i > 0, both of the above cohomology groups vanish. So q_∗f_∗(ω_X⊗ M ) is coherent sheaf generically of rank one, which is a ”cohomological

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principal polarization”. In other words, for all P ∈ Pic⁰(S), the cohomology groups of q_∗f_∗(ωX⊗ M ) ⊗ P are of the same dimension as the cohomology groups of a principal polarization of S.

By [H, Proposition 2.2], there exists an appropriate theta divisor Θ on S and an isomorphism of sheaves q∗f∗(ωX⊗ M ) = OS(Θ). Pulling back with respect to q and f , for all P ∈ Pic⁰(Y ), we get a map of linear series

f^∗q^∗|Θ + P |−−→ |K^×G X+ K⁰+ f^∗P |

which factors through |KX+ M + f^∗P |, where G is a (possibly empty) effective divisor in |K_X+ K⁰− f^∗q^∗Θ|. Since for P ∈ U , the groups H⁰(X, ω_X^⊗2⊗ f^∗P ) are one dimensional, the above map of linear series is an isomorphism, and so G is contained in Bs|K_X+ K⁰+ f^∗P | for all P ∈ U ⊂ Pic⁰(Y ).

Step 3. We wish to show that the divisor G is non-zero and not exceptional with respect to the map X −→ S.

If c = h⁰(X, ω_X(M ) ⊗ Q) = 1. It follows as in Step 2 that there is a theta divisor Θ⁰ on S and effective divisor G⁰ such that

f^∗q^∗|Θ⁰+ P | + G⁰,→ |KX+ K⁰+ Q + f^∗P |, is an injection for all P ∈ Pic⁰(Y ).

If Θ ≡ Θ⁰, then Θ⁰= Θ − P0for some P0∈ Pic⁰(Y ). Replacing Q by Q ⊗ f^∗P0, and P by P ⊗ P₀⁻¹, we conclude that also

f^∗q^∗|Θ + P | + G⁰,→ |KX+ K⁰+ Q + f^∗P |, is an injection for all P ∈ Pic⁰(Y ).

In particular, h⁰(X, G⁰) = h⁰(X, G ⊗ Q) 6= 0. It is easy to see (eg. [Ch, Lemma 7.2]) that G is non-zero and non-exceptional.

We remark now that if D is any ample divisor on S which is not numerically equivalent to Θ, then D^s−i+1· Θⁱ⁻¹> D^s−i· Θⁱ for some 1 ≤ i ≤ s = dimS.

In particular, if Θ 6≡ Θ⁰, there are ample divisors D1, . . . , Ds−1 on S such that Θ⁰· D1· · · Ds−1> Θ · D1· · · Ds−1.

Let H_i be the pull back of D_i and H⁰ be the pull back of an ample divisor on A = Alb(X). We then have the numerical inequality

(KX+ K⁰) · H1· · · Hs−1· (H⁰)^n−s= (KX+ K⁰+ Q) · H1· · · Hs−1· (H⁰)^n−s

≥ (f^∗q^∗Θ⁰) · H1· · · Hs−1· (H⁰)^n−s> (f^∗q^∗Θ) · H1· · · Hs−1· (H⁰)^n−s.

¿From the injection of linear series

f^∗q^∗|Θ + P |−−→ |K^×G _X+ K⁰+ f^∗P |,

one sees that the effective divisor G is not exceptional with respect to the map X −→ S.

If c = h⁰(X, ωX(M ) ⊗ Q) ≥ 2. Let F be a maximal effective divisor such that for all P ∈ Pic⁰(Y ), K_X + K⁰ + Q + f^∗P is linearly equivalent to F + V_P, and h⁰(V_P) ≥ 2.

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We may assume that DP, D⁰_P are two distinct general members of |VP|, which do not contain any components of the ramification divisor of X −→ Y . Therefore there exist divisors ∆P, ∆⁰_P on Y such that modulo X −→ Y exceptional divisors DP = f^∗(∆P) and D_P⁰ = f^∗(∆⁰_P). The map Y −→ S is birational, so there exist effective divisors ¯∆P and ¯∆⁰_P on S and arbitrary q-exceptional divisors EP, E_P⁰ such that

∆P = q^∗( ¯∆P) + EP, ∆⁰_P = q^∗( ¯∆⁰_P) + E_P⁰ .

If h⁰(S, OS( ¯∆P)) = 1, then ¯∆P = ¯∆⁰_P and similarly since Y −→ S is birational,

∆P = ∆⁰_P, which is a contradiction. So ¯∆P is a divisor on S with at least two sections. We may also assume that ¯∆P is an ample divisor on S, since otherwise there would exists a non trivial map of abelian varieties S −→ Σ such that ¯∆_P is the pull back of a divisor on Σ. However then for general P₁, P₂ ∈ Pic⁰(S), VP₁ − VP₂ is rationally equivalent to P1− P2 and is an element of Pic⁰(Σ). The only way this can occur is if S = Σ as required.

Hence, we again obtain the numerical inequality

(KX+ K⁰) · H1· · · Hs−1· (H⁰)^n−s > (f^∗q^∗Θ) · H^s−1· (H⁰)^n−s.

It follows as above that the divisor G is non-zero and not exceptional with respect to the map X −→ S.

Step 4. We show that G being non-exceptional would lead to the desired contradiction. Consider now G0 an irreducible reduced component of G which is not exceptional with respect to the map X −→ S. Since we assumed that h⁰(X, ω_X^⊗2) = 1, it follows that h⁰(X, ωX⊗ P ) = 0 and h⁰(ω_X^⊗2⊗ P ) = 1 for all P ∈ (Pic⁰(Y ) − OY) in an appropriate neighborhood of the origin OS which we again denote by U . From the exact sequence of sheaves

0 −→ ωX⊗ P −→ ωX(G0) ⊗ P −→ ωG₀⊗ P −→ 0, we see that for P ∈ U⁰= U − {O_Y}, there is an isomorphism

H⁰(X, ωX(G0) ⊗ P ) ∼= H⁰(G0, ωG₀⊗ P ).

By Corollary 1.6, there exists a positive dimensional subgroup TG₀ ⊂ Pic⁰(X) such that for all P ∈ TG₀, h⁰(G0, ωG₀⊗ P ) > 0. Since G0 ≺ KX+ K⁰, G0 is vertical with respect to the Iitaka fibration X −→ Y . We may therefore assume that TG₀ is in fact contained in Pic⁰(Y ). It follows that for P ∈ T_G₀∩ U⁰, G₀is not contained in the base locus of the linear series |K_X+ G₀+ P |. Let

|KX+ G0+ P | ^K

0−G0

−−−−→ |KX+ K⁰+ P |

be the map of linear series induced by multiplication by the effective divisor K⁰−G0. Since G₀is in the base locus of |K_X+K⁰+P |, G₀must be an irreducible component of K⁰− G₀. This is the required contradiction since K⁰ is reduced.

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3. Proof of the Main Theorem

Lemma 3.1. Let X and Y be smooth varieties and let A and C be abelian varieties.

Suppose that there is a commutative diagram X −−−−→ A^a

f



 y



 y^π Y −−−−→ C,^q

such that both a and q are generically finite and surjective. Then P_m(Y ) ≤ P_m(X), for all m ≥ 1.

Proof. Let {¯z₁, . . . , ¯z_k} be coordinates on C such that {zi= ¯z_i◦ π | 1 5 i 5 k } may be extended to coordinates {z1, . . . , zn} on A, where n and k are the dimen- sions of A and of C respectively. Let Λ = a^∗dzk+1∧ . . . ∧ a^∗dzn ∈ H⁰(X, Ω^n−k_X ), we claim that

(f^∗ωY)^⊗m ^∧Λ

⊗m

−−−−→ ω_X^⊗m induces the desired injective map

Λm: H⁰(Y, ω^⊗m_Y ) −→ H⁰(X, ω^⊗m_X ).

Since q : Y → C is generically finite and surjective, for a general point y ∈ Y , there are isomorphic Euclidean open neighborhoods Uy and V_q(y) of y and q(y) respectively. For a non-zero section η ∈ H⁰(Y, ω_Y^⊗m), we may assume that η(y) 6= 0.

On Uy, η is represented by ζ · (q^∗d¯z1∧ . . . ∧ q^∗d¯zk)^m, with ζ(y) 6= 0. On f⁻¹(Uy) Λm(η)|_f−1(Uy)= f^∗ζ · (a^∗dz1∧ . . . ∧ a^∗dzn)^m6≡ 0.

Therefore (by choosing harmonic representatives, see [GL2, Lemma 3.1]), Λm : H⁰(Y, ω^⊗m_Y ) −→ H⁰(X, ω_X^⊗m) is injective and so Pm(Y ) ≤ Pm(X).

Theorem 3.2. Let X be a smooth projective variety with P1(X) = P2(X) = 1 and q(X) = dim(X). Then X is birational to an abelian variety.

Proof. Since P₁(X) = P₂(X) and q(X) = dim(X), by Proposition 1.5, it follows that the Albanese morphism is surjective, and hence a generically finite map.

If the Kodaira dimension of X is zero, then by Kawamata’s theorem [Ka1], X is birational to an abelian variety.

Suppose that κ(X) > 0. By Theorem 2.2, the variety X cannot be of general type. Therefore X admits a nontrivial Iitaka fibration. Let f : X → Y be a birational model of the Iitaka fibration. The generic fiber F has κ(F ) = 0 and dim(albX(F )) = dim(F ). Thus albX(F ) is an abelian sub-variety which we denote by K. We obtain a diagram

X −−−−→^alb^X Alb(X)

f



 y



 y^π Y −−−−→ Alb(X)/K^q

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where the morphism g : Y −→ Alb(X)/K is generically finite. By Lemma 3.1, we have P1(Y ) = P2(Y ) = 1. By iterating this procedure, we eventually produce a variety Y⁰ either of maximal Kodaira dimension κ(Y⁰) = dim(Y⁰), or with Kodaira dimension 0 (and hence birational to an abelian variety). By applying Theorem 2.2 or Theorem 2.4, respectively, in conjunction with Lemma 3.1, we conclude that P2(X) ≥ 2.

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Jungkai Alfred Chen, Department of Mathematics, National Chung Cheng Uni- versity, Ming Hsiung, Chia Yi, 621, Taiwan

E-mail address: jkchen@math.ccu.edu.tw

Christopher Derek Hacon, UTAH, Department of Mathematics, Salt Lake City, UT 84112, USA

E-mail address: chhacon@math.utah.edu